Parts of a radical
A radical is an expression with a root symbol.
In ∛27:
| Part | Name |
|---|---|
| ∛ | radical sign |
| 3 | index (cube root) |
| 27 | radicand |
When no index is written, it’s a square root (index 2).
Simplifying square roots
A radical is in simplest radical form when the radicand has no perfect-square factors (other than 1).
Method: Find the largest perfect-square factor and take its root outside.
Example: √50
- 50 = 25 × 2
- √50 = √25 × √2 = 5√2
| Entire radical | Simplified (mixed radical) |
|---|---|
| √12 | 2√3 |
| √18 | 3√2 |
| √48 | 4√3 |
| √72 | 6√2 |
| √200 | 10√2 |
Using prime factorization
Example: √180
- 180 = 2² × 3² × 5
- Pairs come out: 2 × 3 = 6
- 6√5
Cube roots
Take out perfect cubes.
Example: ∛54 = ∛(27 × 2) = 3∛2
Mixed to entire
Reverse the process: square the outside number and multiply it in.
Example: 3√5 = √(9 × 5) = √45
Entire radicals make comparing easy: is 3√5 or 4√3 larger? √45 vs √48 → 4√3.
Radicals with variables
For square roots, each pair of a variable comes out.
- √(x²) = x (for x ≥ 0)
- √(x⁶) = x³
- √(18a³) = √(9a² × 2a) = 3a√(2a)
Approximating
√50 = 5√2 ≈ 5 × 1.414 = 7.07
Simplest radical form is exact; the decimal is an approximation.
Worked examples
Example 1: Simplify √75 → 25 × 3 → 5√3.
Example 2: Simplify √98 → 49 × 2 → 7√2.
Example 3: Write 2∛3 as an entire radical → ∛(8 × 3) = ∛24.
Common mistakes
- Using a perfect square that isn’t the largest: √72 = 2√18 isn’t fully simplified.
- Taking the square root of only part: √(9 + 16) ≠ 3 + 4 (it’s 5).
- Taking out cubes from a square root, or squares from a cube root.
Try it yourself
- √20
- √63
- √128
- ∛40
- Write 5√3 as an entire radical.
Answers: 1) 2√5 2) 3√7 3) 8√2 4) 2∛5 5) √75
Quick recap
- A radical has an index and a radicand.
- Simplify by taking out perfect-square (or perfect-cube) factors.
- Convert between mixed radicals and entire radicals as needed.